Let 1 be the category with one object • and one morphism. Let Δx be the constant functor to x.
A set is a family of • called elements. A set morphism S->S’ has a 1-morphism between each element of S and some element of S’. The 1-morphisms •->Δ•S correspond to the set morphisms Δ∅•->S. The 1-morphisms Δ•S->• correspond to the set morphisms S->Δ1•. We have Δ∅ ⊣ Δ• ⊣ Δ1.
Let 0 the empty category. • is the empty family. A 1-morphism has nothing to prove. There’s no forgetful functor 1->0 so the buck stops here.
Let 1 be the category with one object • and one morphism. Let Δx be the constant functor to x.
A set is a family of • called elements. A set morphism S->S’ has a 1-morphism between each element of S and some element of S’. The 1-morphisms •->Δ•S correspond to the set morphisms Δ∅•->S. The 1-morphisms Δ•S->• correspond to the set morphisms S->Δ1•. We have Δ∅ ⊣ Δ• ⊣ Δ1.
Let 0 the empty category. • is the empty family. A 1-morphism has nothing to prove. There’s no forgetful functor 1->0 so the buck stops here.